Answer
The velocity vector: ${\bf{v}}\left( t \right) = \left( {\frac{1}{2}{t^2} + \frac{1}{3},4t - 2} \right)$
Work Step by Step
Find the velocity vector:
${\bf{v}}\left( t \right) = \smallint {\bf{a}}\left( t \right){\rm{d}}t = \smallint \left( {t,4} \right){\rm{d}}t = \left( {\frac{1}{2}{t^2},4t} \right) + {{\bf{c}}_0}$
The initial condition ${\bf{v}}\left( 0 \right) = \left( {\frac{1}{3}, - 2} \right)$ gives
$\left( {\frac{1}{3}, - 2} \right) = {{\bf{c}}_0}$
Thus,
${\bf{v}}\left( t \right) = \left( {\frac{1}{2}{t^2},4t} \right) + \left( {\frac{1}{3}, - 2} \right)$
${\bf{v}}\left( t \right) = \left( {\frac{1}{2}{t^2} + \frac{1}{3},4t - 2} \right)$
